Identifier
Values
{{1}} => [1] => [1] => [[1],[]] => 1
{{1,2}} => [2,1] => [2] => [[2],[]] => 2
{{1},{2}} => [1,2] => [2] => [[2],[]] => 2
{{1,2,3}} => [2,3,1] => [3] => [[3],[]] => 2
{{1,2},{3}} => [2,1,3] => [3] => [[3],[]] => 2
{{1,3},{2}} => [3,2,1] => [2,1] => [[2,2],[1]] => 3
{{1},{2,3}} => [1,3,2] => [1,2] => [[2,1],[]] => 3
{{1},{2},{3}} => [1,2,3] => [3] => [[3],[]] => 2
{{1,2,3,4}} => [2,3,4,1] => [4] => [[4],[]] => 2
{{1,2,3},{4}} => [2,3,1,4] => [4] => [[4],[]] => 2
{{1,2,4},{3}} => [2,4,3,1] => [3,1] => [[3,3],[2]] => 3
{{1,2},{3,4}} => [2,1,4,3] => [2,2] => [[3,2],[1]] => 4
{{1,2},{3},{4}} => [2,1,3,4] => [4] => [[4],[]] => 2
{{1,3,4},{2}} => [3,2,4,1] => [2,2] => [[3,2],[1]] => 4
{{1,3},{2,4}} => [3,4,1,2] => [4] => [[4],[]] => 2
{{1,3},{2},{4}} => [3,2,1,4] => [2,2] => [[3,2],[1]] => 4
{{1,4},{2,3}} => [4,3,2,1] => [1,2,1] => [[2,2,1],[1]] => 4
{{1},{2,3,4}} => [1,3,4,2] => [1,3] => [[3,1],[]] => 3
{{1},{2,3},{4}} => [1,3,2,4] => [1,3] => [[3,1],[]] => 3
{{1,4},{2},{3}} => [4,2,3,1] => [3,1] => [[3,3],[2]] => 3
{{1},{2,4},{3}} => [1,4,3,2] => [1,2,1] => [[2,2,1],[1]] => 4
{{1},{2},{3,4}} => [1,2,4,3] => [2,2] => [[3,2],[1]] => 4
{{1},{2},{3},{4}} => [1,2,3,4] => [4] => [[4],[]] => 2
{{1,2,3,4,5}} => [2,3,4,5,1] => [5] => [[5],[]] => 2
{{1,2,3,4},{5}} => [2,3,4,1,5] => [5] => [[5],[]] => 2
{{1,2,3,5},{4}} => [2,3,5,4,1] => [4,1] => [[4,4],[3]] => 3
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2] => [[4,3],[2]] => 4
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [5] => [[5],[]] => 2
{{1,2,4,5},{3}} => [2,4,3,5,1] => [3,2] => [[4,3],[2]] => 4
{{1,2,4},{3,5}} => [2,4,5,1,3] => [5] => [[5],[]] => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [3,2] => [[4,3],[2]] => 4
{{1,2,5},{3,4}} => [2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,3] => [[4,2],[1]] => 4
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,3] => [[4,2],[1]] => 4
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [4,1] => [[4,4],[3]] => 3
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]] => 5
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [3,2] => [[4,3],[2]] => 4
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [5] => [[5],[]] => 2
{{1,3,4,5},{2}} => [3,2,4,5,1] => [2,3] => [[4,2],[1]] => 4
{{1,3,4},{2,5}} => [3,5,4,1,2] => [2,3] => [[4,2],[1]] => 4
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,3] => [[4,2],[1]] => 4
{{1,3,5},{2,4}} => [3,4,5,2,1] => [4,1] => [[4,4],[3]] => 3
{{1,3},{2,4,5}} => [3,4,1,5,2] => [3,2] => [[4,3],[2]] => 4
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [5] => [[5],[]] => 2
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [4,1] => [[4,4],[3]] => 3
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [2,1,2] => [[3,2,2],[1,1]] => 4
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [2,3] => [[4,2],[1]] => 4
{{1,4,5},{2,3}} => [4,3,2,5,1] => [1,2,2] => [[3,2,1],[1]] => 5
{{1,4},{2,3,5}} => [4,3,5,1,2] => [1,4] => [[4,1],[]] => 3
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [1,2,2] => [[3,2,1],[1]] => 5
{{1,5},{2,3,4}} => [5,3,4,2,1] => [1,3,1] => [[3,3,1],[2]] => 4
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,4] => [[4,1],[]] => 3
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4] => [[4,1],[]] => 3
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,3,1] => [[3,3,1],[2]] => 4
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]] => 4
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]] => 5
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,4] => [[4,1],[]] => 3
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [3,2] => [[4,3],[2]] => 4
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [3,2] => [[4,3],[2]] => 4
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [2,3] => [[4,2],[1]] => 4
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [3,2] => [[4,3],[2]] => 4
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]] => 4
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]] => 5
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,4] => [[4,1],[]] => 3
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]] => 5
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [2,2,1] => [[3,3,2],[2,1]] => 5
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]] => 4
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [2,3] => [[4,2],[1]] => 4
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [2,3] => [[4,2],[1]] => 4
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [4,1] => [[4,4],[3]] => 3
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]] => 4
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]] => 5
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [3,2] => [[4,3],[2]] => 4
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [5] => [[5],[]] => 2
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Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
Map
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.